• natural orbital functional;
  • reduced density matrices;
  • N-representability;
  • intrapair electron correlation;
  • interpair electron correlation

The natural orbital functional (NOF) theory is briefly reviewed. The meaning of the top-down and bottom-up approaches for the construction of a NOF is analyzed. A particular reconstruction of the two-particle reduced density matrix (2-RDM) based on the cumulant expansion is discussed. The cumulant is expressed by two auxiliary matrices, which are constrained to certain bounds due to the N-representabilty conditions of the 2-RDM. Appropriate forms of these matrices lead to different implementations known in the literature as PNOFi (i = 1–5). The strengths and weaknesses of PNOF5 are assessed. Its main strength is its ability to deal with the intrapair electron correlation at a reasonable computational cost. Its main limitation is the absence of the interpair electron correlation. The inclusion of the missing correlation via a multiconfigurational perturbation theory is shortly described. The growing interest in methods based on NOF theory points to a promising future in this field. © 2014 Wiley Periodicals, Inc.